Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2496,2,Mod(1,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,-2,0,0,0,3,0,-4,0,-3,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.9306603445\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2496.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.622216 q^{5} +4.42864 q^{7} +1.00000 q^{9} -5.80642 q^{11} -1.00000 q^{13} +0.622216 q^{15} +2.00000 q^{17} +4.42864 q^{19} -4.42864 q^{21} -8.85728 q^{23} -4.61285 q^{25} -1.00000 q^{27} -2.00000 q^{29} -7.18421 q^{31} +5.80642 q^{33} -2.75557 q^{35} -0.755569 q^{37} +1.00000 q^{39} +3.37778 q^{41} +7.61285 q^{43} -0.622216 q^{45} +1.80642 q^{47} +12.6128 q^{49} -2.00000 q^{51} -4.75557 q^{53} +3.61285 q^{55} -4.42864 q^{57} -11.0509 q^{59} +8.10171 q^{61} +4.42864 q^{63} +0.622216 q^{65} -8.04149 q^{67} +8.85728 q^{69} -7.05086 q^{71} +7.24443 q^{73} +4.61285 q^{75} -25.7146 q^{77} -12.0000 q^{79} +1.00000 q^{81} +3.05086 q^{83} -1.24443 q^{85} +2.00000 q^{87} -1.86665 q^{89} -4.42864 q^{91} +7.18421 q^{93} -2.75557 q^{95} -0.755569 q^{97} -5.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 2 q^{5} + 3 q^{9} - 4 q^{11} - 3 q^{13} + 2 q^{15} + 6 q^{17} + 13 q^{25} - 3 q^{27} - 6 q^{29} - 8 q^{31} + 4 q^{33} - 8 q^{35} - 2 q^{37} + 3 q^{39} + 10 q^{41} - 4 q^{43} - 2 q^{45} - 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.622216 −0.278263 −0.139132 − 0.990274i \(-0.544431\pi\)
−0.139132 + 0.990274i \(0.544431\pi\)
\(6\) 0 0
\(7\) 4.42864 1.67387 0.836934 − 0.547304i \(-0.184346\pi\)
0.836934 + 0.547304i \(0.184346\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.80642 −1.75070 −0.875351 − 0.483487i \(-0.839370\pi\)
−0.875351 + 0.483487i \(0.839370\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0.622216 0.160655
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.42864 1.01600 0.508000 − 0.861357i \(-0.330385\pi\)
0.508000 + 0.861357i \(0.330385\pi\)
\(20\) 0 0
\(21\) −4.42864 −0.966408
\(22\) 0 0
\(23\) −8.85728 −1.84687 −0.923435 − 0.383754i \(-0.874631\pi\)
−0.923435 + 0.383754i \(0.874631\pi\)
\(24\) 0 0
\(25\) −4.61285 −0.922570
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −7.18421 −1.29032 −0.645161 − 0.764047i \(-0.723210\pi\)
−0.645161 + 0.764047i \(0.723210\pi\)
\(32\) 0 0
\(33\) 5.80642 1.01077
\(34\) 0 0
\(35\) −2.75557 −0.465776
\(36\) 0 0
\(37\) −0.755569 −0.124215 −0.0621074 − 0.998069i \(-0.519782\pi\)
−0.0621074 + 0.998069i \(0.519782\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.37778 0.527521 0.263761 − 0.964588i \(-0.415037\pi\)
0.263761 + 0.964588i \(0.415037\pi\)
\(42\) 0 0
\(43\) 7.61285 1.16095 0.580474 − 0.814279i \(-0.302867\pi\)
0.580474 + 0.814279i \(0.302867\pi\)
\(44\) 0 0
\(45\) −0.622216 −0.0927544
\(46\) 0 0
\(47\) 1.80642 0.263494 0.131747 − 0.991283i \(-0.457941\pi\)
0.131747 + 0.991283i \(0.457941\pi\)
\(48\) 0 0
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −4.75557 −0.653228 −0.326614 − 0.945158i \(-0.605908\pi\)
−0.326614 + 0.945158i \(0.605908\pi\)
\(54\) 0 0
\(55\) 3.61285 0.487156
\(56\) 0 0
\(57\) −4.42864 −0.586588
\(58\) 0 0
\(59\) −11.0509 −1.43870 −0.719349 − 0.694648i \(-0.755560\pi\)
−0.719349 + 0.694648i \(0.755560\pi\)
\(60\) 0 0
\(61\) 8.10171 1.03732 0.518659 − 0.854981i \(-0.326431\pi\)
0.518659 + 0.854981i \(0.326431\pi\)
\(62\) 0 0
\(63\) 4.42864 0.557956
\(64\) 0 0
\(65\) 0.622216 0.0771764
\(66\) 0 0
\(67\) −8.04149 −0.982424 −0.491212 − 0.871040i \(-0.663446\pi\)
−0.491212 + 0.871040i \(0.663446\pi\)
\(68\) 0 0
\(69\) 8.85728 1.06629
\(70\) 0 0
\(71\) −7.05086 −0.836783 −0.418391 − 0.908267i \(-0.637406\pi\)
−0.418391 + 0.908267i \(0.637406\pi\)
\(72\) 0 0
\(73\) 7.24443 0.847897 0.423948 − 0.905686i \(-0.360644\pi\)
0.423948 + 0.905686i \(0.360644\pi\)
\(74\) 0 0
\(75\) 4.61285 0.532646
\(76\) 0 0
\(77\) −25.7146 −2.93045
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.05086 0.334875 0.167437 − 0.985883i \(-0.446451\pi\)
0.167437 + 0.985883i \(0.446451\pi\)
\(84\) 0 0
\(85\) −1.24443 −0.134978
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −1.86665 −0.197864 −0.0989321 − 0.995094i \(-0.531543\pi\)
−0.0989321 + 0.995094i \(0.531543\pi\)
\(90\) 0 0
\(91\) −4.42864 −0.464248
\(92\) 0 0
\(93\) 7.18421 0.744968
\(94\) 0 0
\(95\) −2.75557 −0.282715
\(96\) 0 0
\(97\) −0.755569 −0.0767164 −0.0383582 − 0.999264i \(-0.512213\pi\)
−0.0383582 + 0.999264i \(0.512213\pi\)
\(98\) 0 0
\(99\) −5.80642 −0.583568
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2496.2.a.bk.1.2 3
3.2 odd 2 7488.2.a.cy.1.2 3
4.3 odd 2 2496.2.a.bl.1.2 3
8.3 odd 2 1248.2.a.o.1.2 ✓ 3
8.5 even 2 1248.2.a.p.1.2 yes 3
12.11 even 2 7488.2.a.cx.1.2 3
24.5 odd 2 3744.2.a.z.1.2 3
24.11 even 2 3744.2.a.ba.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.o.1.2 ✓ 3 8.3 odd 2
1248.2.a.p.1.2 yes 3 8.5 even 2
2496.2.a.bk.1.2 3 1.1 even 1 trivial
2496.2.a.bl.1.2 3 4.3 odd 2
3744.2.a.z.1.2 3 24.5 odd 2
3744.2.a.ba.1.2 3 24.11 even 2
7488.2.a.cx.1.2 3 12.11 even 2
7488.2.a.cy.1.2 3 3.2 odd 2